🤖 AI Summary
This study addresses the challenge of effectively classifying graphs with heterogeneous structures and sizes without relying on node embeddings or graph alignment. To this end, the authors extend the k-nearest neighbors (kNN) classifier to metric spaces induced by the Gromov–Wasserstein (GW) and fused Gromov–Wasserstein (fGW) distances, treating graphs as finite metric measure spaces equipped with node features. The work establishes, for the first time, the universal consistency of both GW-kNN and fGW-kNN classifiers over the space of graphs and their attribute-augmented extensions, thereby providing a rigorous theoretical foundation for nonparametric classification based on GW-type distances. Empirical evaluations demonstrate that the proposed approach achieves strong performance and excellent generalization across multiple graph benchmark datasets.
📝 Abstract
The Gromov--Wasserstein (GW) distance provides a framework for comparing metric measure spaces, regardless of their underlying structure or geometry. For network-based data, it enables direct comparisons of graphs with different numbers of nodes, without requiring an embedding or other abstraction. Furthermore, through a variant of GW known as fused Gromov--Wasserstein (fGW), it is also possible to incorporate node features in addition to graph structure. In this work, we implement $k$-nearest neighbors ($k$-NN) classification using the GW and fGW distances. We prove the universal consistency of the GW-$k$-NN classifier on the space of equivalence classes of metric measure spaces with finite support and uniform probability measure. By viewing graphs as finitely supported metric measure spaces equipped with the pairwise distance metric and a uniform probability measure on the nodes, we obtain universal consistency of GW-$k$-NN for the space of graphs. Likewise for fGW-$k$-NN, we prove universal consistency on the space of weak isomorphism classes of structured objects consisting of metric measure spaces with finite support and uniform probability measure and feature maps into Euclidean space, thus establishing universal consistency on the space of node-attributed graphs. Our numerical experiments show that GW-$k$-NN and fGW-$k$-NN consistently perform well across multiple graph datasets, suggesting that metric classifiers such as $k$-NN work well in the GW framework.